C₆₁×#13500 #29516933

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#13500

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#2076
prime radical	C₂×#41×C₆₁	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#131×C₆₁ ◁ G

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 125 points:

(1,2,6,11,26,40,57,53,7,12,29,43,5,10,25,39)(3,13,16,45,15,34,50,41,18,33,27,61,17,8,30,32)(4,19,22,38,24,59,60,37,23,54,58,36,21,42,28,9)(14,46,48,31,47,52,62,63,20,55,56,35,49,64,51,44)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3)(2,8)(4,20)(5,17)(6,27)(7,18)(9,35)(10,33)(11,41)(12,34)(13,40)(14,23)(15,26)(16,29)(19,52)(21,47)(22,48)(24,49)(25,50)(28,51)(30,57)(31,37)(32,43)(36,63)(38,44)(39,45)(42,46)(53,61)(54,64)(55,59)(56,58)(60,62);(1,4,7,23)(2,9,12,37)(3,14)(5,24,26,21)(6,28,29,60)(8,31)(10,38,40,36)(11,42,43,59)(13,44)(15,49)(16,51)(17,47)(18,20)(19,53,54,39)(22,57,58,25)(27,62)(30,48)(32,46)(33,63)(34,35)(41,55)(45,64)(50,56)(52,61)

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6488);